From a0a5f0f799be512684e31d13cdedc2ef1c404792 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Andreas=20M=C3=BCller?= <andreas.mueller@ost.ch>
Date: Sat, 13 Mar 2021 21:09:35 +0100
Subject: add additional paper

---
 vorlesungen/slides/8/markov/irreduzibel.tex | 136 ++++++++++++++++++++++++++++
 1 file changed, 136 insertions(+)
 create mode 100644 vorlesungen/slides/8/markov/irreduzibel.tex

(limited to 'vorlesungen/slides/8/markov/irreduzibel.tex')

diff --git a/vorlesungen/slides/8/markov/irreduzibel.tex b/vorlesungen/slides/8/markov/irreduzibel.tex
new file mode 100644
index 0000000..87e90e4
--- /dev/null
+++ b/vorlesungen/slides/8/markov/irreduzibel.tex
@@ -0,0 +1,136 @@
+%
+% irreduzibel.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Irreduzible Markovkette}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\r{2}
+\coordinate (A) at ({\r*cos(0*60)},{\r*sin(0*60)});
+\coordinate (B) at ({\r*cos(1*60)},{\r*sin(1*60)});
+\coordinate (C) at ({\r*cos(2*60)},{\r*sin(2*60)});
+\coordinate (D) at ({\r*cos(3*60)},{\r*sin(3*60)});
+\coordinate (E) at ({\r*cos(4*60)},{\r*sin(4*60)});
+\coordinate (F) at ({\r*cos(5*60)},{\r*sin(5*60)});
+
+\uncover<-2>{
+\draw (A) -- (B);
+\draw (A) -- (C);
+\draw (A) -- (D);
+\draw (A) -- (E);
+\draw (A) -- (F);
+
+\draw (B) -- (A);
+\draw (B) -- (C);
+\draw (B) -- (D);
+\draw (B) -- (E);
+\draw (B) -- (F);
+
+\draw (C) -- (A);
+\draw (C) -- (B);
+\draw (C) -- (D);
+\draw (C) -- (E);
+\draw (C) -- (F);
+
+\draw (D) -- (A);
+\draw (D) -- (B);
+\draw (D) -- (C);
+\draw (D) -- (E);
+\draw (D) -- (F);
+
+\draw (E) -- (A);
+\draw (E) -- (B);
+\draw (E) -- (C);
+\draw (E) -- (D);
+\draw (E) -- (F);
+
+\draw (F) -- (A);
+\draw (F) -- (B);
+\draw (F) -- (C);
+\draw (F) -- (D);
+\draw (F) -- (E);
+}
+
+\uncover<3->{
+
+\draw[->,color=black!30,shorten >= 0.15cm,line width=3pt] (A) to[out=90,in=-30] (B);
+\draw[->,color=black!70,shorten >= 0.15cm,line width=3pt] (A) -- (C);
+\draw[->,color=black!20,shorten >= 0.15cm,line width=3pt] (B) -- (A);
+\draw[->,color=black!60,shorten >= 0.15cm,line width=3pt] (B) to[out=150,in=30] (C);
+\draw[->,color=black!20,shorten >= 0.15cm,line width=3pt] (B) to[out=-90,in=-150,distance=1cm] (B);
+\draw[->,color=black!50,shorten >= 0.15cm,line width=3pt] (C) to[out=-60,in=180] (A);
+\draw[->,color=black!50,shorten >= 0.15cm,line width=3pt] (C) -- (B);
+
+\draw[->,color=black!40,shorten >= 0.15cm,line width=3pt]
+	(D) to[out=-90,in=150] (E);
+\draw[->,color=black!30,shorten >= 0.15cm,line width=3pt]
+	(E) -- (D);
+\draw[->,color=black!70,shorten >= 0.15cm,line width=3pt]
+	(E) to[out=-30,in=-150] (F);
+\draw[->,color=black!40,shorten >= 0.15cm,line width=3pt]
+	(F) -- (E);
+\draw[->,color=black!60,shorten >= 0.15cm,line width=3pt]
+	(F) to[out=120,in=0] (D);
+\draw[->,color=black!60,shorten >= 0.15cm,line width=3pt]
+	(D) -- (F);
+}
+
+\fill[color=white] (A) circle[radius=0.2];
+\fill[color=white] (B) circle[radius=0.2];
+\fill[color=white] (C) circle[radius=0.2];
+\fill[color=white] (D) circle[radius=0.2];
+\fill[color=white] (E) circle[radius=0.2];
+\fill[color=white] (F) circle[radius=0.2];
+
+\draw (A) circle[radius=0.2];
+\draw (B) circle[radius=0.2];
+\draw (C) circle[radius=0.2];
+\draw (D) circle[radius=0.2];
+\draw (E) circle[radius=0.2];
+\draw (F) circle[radius=0.2];
+
+\node at (A) {$1$};
+\node at (B) {$2$};
+\node at (C) {$3$};
+\node at (D) {$4$};
+\node at (E) {$5$};
+\node at (F) {$6$};
+
+\end{tikzpicture}
+\end{center}
+\uncover<2->{%
+\begin{block}{Irreduzibel}
+Graph zusammenhängend $\Rightarrow$
+Keine Zerlegung in Teilgraphen möglich
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{Reduzibel}
+Die Zustandsmenge zerfällt in zwei disjunkte Teilmengen $V=V_1\cup V_2$
+und es gibt keine Übergängen zwischen den Mengen:
+\uncover<4->{%
+\begin{align*}
+P
+&=
+\begin{pmatrix*}[l]
+0  &0.2&0.5&   &   &   \\
+0.3&0.2&0.5&   &   &   \\
+0.7&0.6&0  &   &   &   \\
+   &   &   &0  &0.3&0.4\\
+   &   &   &0.4&0  &0.6\\
+   &   &   &0.6&0.7&0  
+\end{pmatrix*}
+\end{align*}}%
+\uncover<5->{%
+$P$ zerfällt in zwei Blöcke die unabhängig voneinander analysiert werden können
+}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
-- 
cgit v1.2.1